In a book I am reading the author states an $L^\infty$ bound for a function which is meant to follow from its Lipschitz constant, i.e., if for all $x,y\in B\subset\mathbb{R}^d$, $$|f(x)-f(y)|\leq M|x-y|,$$ then $||f||_{L^\infty(B)}\leq M$?
I am not able to prove this. The standard method would be to write $|f(x)|\leq |f(x)-f(x_0)|+|f(x_0)|\leq M~diam(B)+|f(x_0)|$. However, this does not match the simple bound which is claimed in the book.
Is the claim false? or is it standard to estimate bounds on a function in terms of its modulus of continuity?
Without more information on $B$ and/or on $M$, it's not correct. Take $f(x)=x$ and $B=[0,2]$. Then $|f(x)-f(y)|\leq |x-y|$ but $\|f\|_{L^\infty (B)}=2>1.$